A cylindrical container for storing radioactive waste is to be constructed from lead and have a thickness of 6 in.(see the figure).If the volume of the outside cylinder is to be , find the radius and the height of the inside cylinder that will result in a container of maximum storage capacity. Hint: Show that the storage capacity (inside volume) is given by
A)r = ft.; h = 2 ft.
B)r = ft.; h = 3 ft.
C)r = ft.; h = 2 ft.
D)r = ft.; h = 3 ft.

Question

A cylindrical container for storing radioactive waste is to be constructed from lead and have a thickness of 6 in.(see the figure).If the volume of the outside cylinder is to be  , find the radius and the height of the inside cylinder that will result in a container of maximum storage capacity.  Hint: Show that the storage capacity (inside volume) is given by 
A)r = ft.; h = 2 ft. 
B)r = ft.; h = 3 ft. 
C)r = ft.; h = 2 ft. 
D)r = ft.; h = 3 ft.

Quizplus · Accepted Answer

We are trying to maximize the storage capacity of the container, which is given by:

V = πh(R^2 - r^2)

where h is the height of the inside cylinder, R is the outer radius of the container (which is equal to r + 6 in this case), and r is the radius of the inside cylinder.

We know that the volume of the outside cylinder is given as 11eb12b0_a9df_267e_b394_f36cb24b9140_TB7866_11, which means:

11eb12b0_a9df_267e_b394_f36cb24b9140_TB7866_11 = π(R^2)(6)

Solving this for R, we get:

R = 11eb12b0_a9df_4d8f_b394_642bd54e3f2a_TB7866_11 in

Substituting this into the formula for V, we get:

V = πh[(11eb12b0_a9df_4d8f_b394_642bd54e3f2a_TB7866_11)^2 - r^2]

To find the maximum V, we can take the derivative of this formula with respect to r, set it equal to zero, and solve for r:

dV/dr = πh(-2r) = 0

r = 0

or

h = 11eb12b0_a9df_4d90_b394_7344746c404d_TB7866_11/2π

We can't have r = 0, so we take the second option:

r = 11eb12b0_a9df_4d90_b394_7344746c404d_TB7866_11/2π ≈ 1.75 in

Now we need to find h to maximize V. We know that the total height of the container is h + 12 (6 inches on top and bottom), so we write:

h + 12 = H

where H is a constant. Solving this for h, we get:

h = H - 12

Substituting this into the formula for V, we get:

V = π(H - 12)[(11eb12b0_a9df_4d8f_b394_642bd54e3f2a_TB7866_11)^2 - (11eb12b0_a9df_4d90_b394_7344746c404d_TB7866_11/2π)^2]

To find the maximum V, we can take the derivative of this formula with respect to H, set it equal to zero, and solve for H:

dV/dH = π[(11eb12b0_a9df_4d8f_b394_642bd54e3f2a_TB7866_11)^2 - (11eb12b0_a9df_4d90_b394_7344746c404d_TB7866_11/2π)^2] = 0

Solving this, we get:

H = 12 + 2(11eb12b0_a9df_4d90_b394_7344746c404d_TB7866_11/π) ≈ 27.5 in

Therefore, the radius and height of the inside cylinder that will result in a container of maximum storage capacity is:

r ≈ 1.75 in
h = H - 12 ≈ 15.5 in

Converting to feet, we get:

r ≈ 0.146 ft
h ≈ 1.292 ft

So the answer is choice B.

A Cylindrical Container for Storing Radioactive Waste Is to Be